| L(s) = 1 | + (−0.347 − 1.96i)2-s + (−3.75 + 1.36i)4-s + (−3.46 − 2.91i)5-s + (22.9 + 8.37i)7-s + (4 + 6.92i)8-s + (−4.52 + 7.84i)10-s + (24.6 − 20.7i)11-s + (0.761 − 4.31i)13-s + (8.49 − 48.2i)14-s + (12.2 − 10.2i)16-s + (16.1 − 28.0i)17-s + (−62.8 − 108. i)19-s + (17.0 + 6.19i)20-s + (−49.3 − 41.4i)22-s + (138. − 50.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (−0.310 − 0.260i)5-s + (1.24 + 0.451i)7-s + (0.176 + 0.306i)8-s + (−0.143 + 0.247i)10-s + (0.676 − 0.567i)11-s + (0.0162 − 0.0920i)13-s + (0.162 − 0.920i)14-s + (0.191 − 0.160i)16-s + (0.231 − 0.400i)17-s + (−0.758 − 1.31i)19-s + (0.190 + 0.0692i)20-s + (−0.478 − 0.401i)22-s + (1.25 − 0.456i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.19896 - 1.08082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19896 - 1.08082i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.46 + 2.91i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-22.9 - 8.37i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (-24.6 + 20.7i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-0.761 + 4.31i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-16.1 + 28.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (62.8 + 108. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-138. + 50.4i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-21.0 - 119. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-148. + 54.0i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (151. - 262. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-80.0 + 453. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-121. + 101. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (99.9 + 36.3i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 91.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-251. - 211. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-592. - 215. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (158. - 900. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (534. - 926. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-371. - 643. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (188. + 1.07e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-128. - 731. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (587. + 1.01e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-376. + 315. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87482601739411572447878666657, −11.36221137384873132738855430993, −10.37085604957108935959203940848, −8.803363878098552385088352261959, −8.521192052037481281976115474461, −6.97810910423050299645843866606, −5.27165207670530244648346047417, −4.27667219344017000567861216773, −2.59677922167911824871823506443, −0.939365214276196892920983060250,
1.48098852035221266761101956755, 3.87301523684242734075358109291, 4.95044602521977667541286800127, 6.36711984868147547006350080397, 7.51775065812314380840728407057, 8.219851779565957261367548673397, 9.461671094769434407275618109486, 10.64478322617307036200553700893, 11.54850044730896804905946845246, 12.68284259657268631637307674512
